Variational analysis of Nash equilibria for a model of traffic flow
Alberto Bressan, Chen Jie Liu, Wen Shen and Fang Yu
Abstract: The paper is concerned with Nash equilibrium solutions for the Lighthill–Whitham model of traffic flow, where each driver chooses his own departure time in order to minimize the sum of a departure cost and an arrival cost. Estimates are provided, on how much the Nash solution may change, depending on the cost functions and on the flux function of the conservation law. It is shown that this equilibrium solution can also be determined as a global minimizer for a functional Φ, measuring the maximum total cost among all drivers, in a given traffic pattern. The last section of the paper introduces two evolution models, describing how the traffic pattern can change, day after day. It is assumed that each driver adjusts his departure time based on previous experience, in order to lower his own cost. Numerical simulations are reported, indicating a possible instability of the Nash equilibrium.